Question

Find one irrational number between $ 0.2101 $ and $ 0.2222 \ldots $ .
(A) $ 0.2 $
(B) $ 0.25 $
(C) $ 0.22010010001 \ldots $
(D) None of these

Answer 1

Hint: In this problem, to find one irrational number between given decimal expansions we will observe their digits. Remember that the decimal expansion of an irrational number is non-terminating non-recurring.

Complete step-by-step answer:
In the given problem, we have to find one irrational number between the numbers $ 0.2101 $ and $ 0.2222 \ldots $ . We can write $ 0.\bar 2 = 0.2222 \ldots $ . Note that $ 0.2101 $ is called terminating decimal expansion and $ 0.2222 \ldots $ is called non-terminating recurring (repeating) decimal expansion.
Let us assume that $ a = 0.2101 $ and $ b = 0.2222 \ldots $ . To find one irrational number between $ a $ and $ b $ , let us observe the digits in their decimal expansions (representations). So, we can say that the first place (digit) in their decimal expansions are equal. That is, the first digit is $ 2 $ . Hence, the first digit of decimal expansion of the required irrational number will be $ 2 $ .
Now observe the second place so we can say that the second place in their decimal expansions are distinct (different). That is, in the decimal expansion of number $ a $ the second digit is $ 1 $ and in the decimal expansion of number $ b $ the second digit is $ 2 $ . Now we can say that the number $ a $ is less than the number $ b $ . Hence, the second digit of decimal expansion of the required irrational number will be $ 2 $ .
Now observe the third place so we can say that the third place in their decimal expansions are distinct (different). That is, in the decimal expansion of number $ a $ the third digit is $ 0 $ and in the decimal expansion of number $ b $ the third digit is $ 2 $ . Hence, the third digit of decimal expansion of the required irrational number will be either $ 0 $ or $ 1 $ because $ a < b $ .
Hence, the required irrational number is $ 0.220 \ldots $ or $ 0.221 \ldots $ between $ 0.2101 $ and $ 0.2222 \ldots $ . That is, the required irrational number is $ 0.22010010001 \ldots $ . Hence, option C is correct.
So, the correct answer is “Option C”.

Note: In the given problem, directly we can say that option A and B are wrong because they are decimal expansions of rational numbers and we have to find irrational numbers. Remember that the decimal expansion of a rational number is either terminating or non-terminating recurring.